I'm becoming more interested in Marxism, and the ensuing economies that attempted to use Marxist thought. What I wonder is, which post-Marx, modern thinkers have most successfully built on Marxist thought?
EDIT: I no longer stand by this answer. Marxism is much more lively, and viable, than I believed in 2017. It has many strands and adherents in various fields. My thoughts on the labor theory of value are simply incorrect, based on a silly misunderstanding. I doubt I shall come back here to fix things; if I do, I will remove this notice. In the meantime I'm turning this answer into a community wiki, in the hopes that others with more time and interest can improve it.
Off the top of my head there are probably four* lively strands of Marxist thought, concerning, respectively, economic matters, identity politics, politics generally, and ideology. (All the thinkers I cite here are of course broad and on a deep level it doesn't make much sense to categorize them as I have. This is just a jumping-off point.)
We shouldn't consider the purely economic strain of Marxist thought because it does indeed seem to have been debunked. The labor theory of value is simply false, so his prophecies have not come to pass. The revolution was supposed to be a historical necessity, even if not a historical inevitability, and yet while it has not happened, other world-historical events have. Plus workers are not necessarily paid merely what they need to survive. We (well, I) no longer live in industrial, but rather post-industrial nations, and Marx could not possibly have foreseen post-industrialism. Wikipedia seems to think the last big thing in the field was done by Kautsky, so my feelings are perhaps not uncommon.
I highly recommend pursuing intersectional studies as well as decolonization. For the latter Frantz Fanon is particularly well regarded - The Wretched of the Earth is a good place to start. For the former, Simone de Beauvoir's Second Sex is rightfully a classic of feminist philosophy. It isn't strictly speaking Marxist but certainly builds on Marxist logic, e.g., the way material circumstances shape ideas - in this case, woman's oppression being internalized. Her work has been much improved-upon. I'd recommend, to start, Judith Butler and Patricia Collins. Collins' Black Feminist Thought is an ideal introduction to third-wave, intersectional feminism. (See also this.) Butler is a dense, maybe inaccessible writer who owes a lot to Derrida. Her main claim is that gender is performative---see Gender Trouble.
Politics: Badiou is a Maoist and metaphysician who tries to tie the two together. His main work is Being and Event, but it's pretty dense and uses a lot of set theory. If you just want to get a handle on how he thinks Manifesto for Philosophy is probably a better call. Toni Negri (with Michael Hardt) wrote the landmark Empire, which draws on a lot of the work done by others on this list. It's about modern-day imperialism and contains a strong critique of dialectics and mediation as tools for thought.
And finally we have ideology/culture generally. Gramsci's Prison Notebooks are on this front far ahead of their time. Adorno and Althusser are chronologically the next step, but I haven't read too much of either. Supposedly "Ideology and Ideological State Apparatuses" is quite good, and that's by Althusser. As for Adorno (with Horkheimer), you'll want Dialectic of Enlightenment, though you have to take the bits on jazz with a grain of salt.
Finally there are Foucault and Zizek. It's hard to know what Foucault's actual political beliefs are but he is entirely in debt to Marx and Nietzsche for his genealogical account of oppression. No matter what your interests are you should read Discipline and Punish. Zizek is controversial because he's still alive and sometimes he says really tremendously stupid things but I found The Sublime Object of Ideology to be pretty good. What's really accessible, and in fact down right funny, are his movies: Pervert's Guide to Cinema and Pervert's Guide to Ideology. They're a very fun jumping-off point.
*There's also the whole world-spirit dialectical-materialism history-as-a-story thing which I find a bit far-fetched, but if you want to go down that road there's always Marcuse with Eros and Civilization.
1) Antonio Negri is probably the person that has most radically built on Marxist thought in a way that attempts to explore the philosophical implications of the changes the world has seen since the 19th century. Part of what's most radical about his project has been the de-Hegelianization of Marx's thought with him. His is probably the most profound reengagement Marx's work since Althusser.
2) David Harvey is a geographer by profession but is one of the most renown contemporary Marxists that attempts to analyze the world today from a Marxist lense. He isn't a philosopher so his language and work would likely be more "accessible" than that of Negri who is a bit more difficult to digest if you're not familiar with others that have heavily influenced him, like Spinoza.
3) Terry Eagleton & Frederic Jameson might be good to look at as well as they both approach contemporary Marxism from its cultural aspects as opposed to the economics side that Harvey would be more versed in or the philosophical that Negri would be the best source on. Jacques Ranciere might also be considered in this category, but Ranciere is a fairly adept philosopher.
4) Surrounding Negri are other Marxist theorists that are interesting in their own right: Yann Mouliere Boutang is an economist by profession who's specialty these days are understanding the cutting edge of capitalist development from a Marxist standpoint (that is to say, to analyze the political implication of the broad scale evolutions of Capital); and Christian Marazzi is similar to Boutang, but focuses more on the financial/monetary side of exploring contemporary capitalism and the political and philosophical implications of things like wage relations/exploitation etc.
5) Etienne Balibar since the 1960's has been cranking out analyses of Marx's work and attempting to reinterpret his questions and methodology in novel ways. He is not a very flashy philosopher but he is very thorough and thoughtful. He engages with the same types of subjects as a Badiou or Zizek but he's keeps the discussion at much more practical level.
Trotsky's theory of Permanent Revolution.
Roman Rosdolsky. A very in depth analysis of Das Kapital and its relations to the Grundrisse and Theories of Surplus Value.
Wilhelm Reich. A failed but still very radical and interesting attempt to bring together Marx and Freud's contributions.
Piero Sraffa. A definitive refutation of the hocus pocus school of marginalist economics.
Moshe Postone, the Krisis/Exit groups, particularly Robert Kurz and Roswitha Scholz. The most radical reading of Marx that I know of, albeit probably unilateral and leading to a political cul-de-sac.
Several English Marxist historians, most prominently (although many times in opposite ways) Perry Anderson and E. P. Thompson.