Update: I figured out the chunk loading error. It's a client-side issue that is exacerbated by minuscule network latency.
As it turns out, Minecraft 1.5 introduced an "advanced OpenGL" option which allows the client to only render those blocks that it can see. Unfortunately, it appears that this also affects the chunks that are loaded from the server. As soon as you disable advanced OpenGL, the chunk loading behavior returns to normal even over the network. Advanced OpenGL is disabled by default, but some users may be provoked into thinking it's a good idea to enable it if they read the patch notes (reading? what's that?).
What I believe is happening is that, with the advanced OpenGL support enabled, the client knows which chunks it can and cannot render based on visibility. As the player moves around the world, the slight network latency between the client and server is enough to delay chunk rendering (or rather, retrieval) from the server long enough to be noticeable. The more you more, the more chunks get loaded into the queue to retrieve from the server and the more noticeable the chunk errors become. By disabling this feature, Minecraft returns to its default behavior of grabbing all nearby chunks.
It does appear that there are still some minor chunk-related issues, but I'm not entirely certain if it's my client or the new server software. Approaching any unloaded chunks will force them to load, however.
TL;DR: Disable Advanced OpenGL support from Options -> Video settings if you're encountering chunk-related loading issues.
The server software has been updated to Minecraft 1.5. There are some minor issues related to chunk loading errors (again), so further testing is needed. It seems that once a number of chunks have been loaded into memory, the errors will slowly diminish. This seems to occur with vanilla Minecraft and Bukkit, so I'm going to test this on my local server.
Be sure to create a backup of your Minecraft install prior to updating (under %APPDATA%/Roaming/.minecraft). If the chunk loading issues are severe, we may stick with 1.4 for the time being. If you lost the backup of your Minecraft install because you updated, well, the TOS prohibits us from distributing the Minecraft .jars. You may need to use MCNostalgia instead.