Alternative ways to clear the Drupal cache
While your cache bins are in MySQL, truncating tables is by far and away the quickest method to clear the cache. It's not a particularly expensive operation, so if it's backing up your db connections I would get your sysadmin to investigate what config issues might be causing that (or whether it's simply a lack of sufficient resources).

As MPD notes in the comments above, your real problem might actually be the cache rebuild after the flush has happened, though. You should be able to confirm that after a bit of testing.

You can take the pressure off the database by using redis/memcache/etc as a cache backend instead of the database. Doing that will also make considerable speed improvements to your site in general.

Your mileage will vary, of course, but it's very likely that flushing cache from memory will be a quicker operation than flushing it from the disk. As Mołot notes below, writing the new cache items to memory rather than disk will also give you a performance boost (there will still be reads and writes to MySQL, but far fewer writes in theory).

As with anything, it needs testing on the bare metal to see how any of this would affect your specific configuration.