There are two main views on the semantics of propositions: tensed vs tenseless. Quoting Markosian:
Tensed view: Propositions have truth values at times rather than just having truth values simpliciter. [This makes it] possible for a proposition to have different truth values at different times.
Tenseless view: Propositions have truth values simpliciter rather than having truth values at times. [This makes it impossible] for a proposition to have different truth values at different times.
The 'modern conception' that Kenny is referring to is the one inherited from mathematics, where sentences don't include such locutions as 'I', 'now', 'yesterday' and so on, and thus the need to make sentences temporally unstable (to use Rescher's phrase) doesn't arise.
The tenseless view of propositions is still prevalent in the mathematical practice, but it seems a little too strong to claim that it's the modern conception of the semantics of propositions. The alternatives are out there, and philosophers of time, logicians interested in the temporal modalities, and others in linguistics and computer science departments are debating the pros and cons of each view. So yes, there is more than one modern conception of the semantics of propositions.
Further Reading. I would recommend reading the aforementioned SEP article on Time in its entirety. If you're interested in a more formal treatment you can look at van Benthem's Modal Logic for Open Minds, chapter 18, and/or Goldblatt's Logics of Time and Computation, chapter 6, both from CSLI.