Chain skipping on new freewheel and chain setup - What went wrong? BACKGROUND NOTE: The bike is a 1992 Univega MTB that originally came with 6-speed SIS with a triple chainring. I removed the SIS shifter when it broke, and went with a friction shifter.  The bike is used as an all-purpose commuter/errands bike; I was having no problems with the "old-tech" setup other than a worn chain, and I have no plans to change the friction shifter on an old bike.

OLD SETUP 
6-speed Sunrace Freewheel (straight-cut, 14-28); 6-speed chain (don't know the type).

NEW SETUP
7-speed DNP Epoch freewheel (beveled or ramped cut cogs, 11,13,15,18,21,24,32); KMC Z50 chain (1/2" x 3/32", 6-7 speed). The KMC chain box says "Indexed."

BOTH SETUPS USE:
Long-cage rear derailleur (Shimano Exage), triple chainring (28,38,48; Shimano Alivio, 18-24 months), and friction shifters.

THE REASON for the change: 
chain wear and skipping, wanting two faster cogs and a lower hill-climber cog. The new chain was adjusted for length using the chain adjustment equation on Parktools web site.  After installing the chain, I doubled-checked and eliminated any chain kinking.  The rear derailleur is adjusted and shifting well with the new setup. When I re-installed the rear wheel, I placed a thin washer between the right-side axle nut and the dropout for chain clearance on the 11-cog but it might not be necessary.

THE PROBLEM: 
The new chain is intermittently skipping in a "skip-forward" or half-shift" motion; less so when riding casually, more so when riding under load; less so on the 48-tooth chainring, appears to be more so on the smaller chain rings.

What's wrong with my new setup?
sounds like your chainrings are worn.  Look at the rings and see if they are "shark fin" shaped rather than symmetrical - a google image search for "worn out chainring" will show examples including some really, really worn out rings.  The old worn out chain likely wore out the rings as well.

I've had exactly what you're describing on a couple bikes and it was remedied with a new chainring up front.