Finally found the setting that makes my icons stay put. It's in the KDE Control Center's "KDE Components" --> "Session Manager" section, under "On Login". Click the "Restore manually saved session" radio button, and KDE will put a "Save Session" icon on your K menu. After rearranging your desktop to your liking, click the "Save Session" icon. Next time KDE starts up, everything should be as you left it (including running programs).

(This assumes you've done a hard disk install, of course.)