Chain skipping under increased pedal torque I read the related post, but my situation is a bit different.  Additional background - I installed new 105 shifters and derailleurs two years ago (~5000 miles) along with chain and cassette.  I also had a 53T chain ring put on in 2013.  All was good.  Had slight hang-up with slack shifting (to higher gears) on rear cogs this spring, so I replaced the cable and sheaths to the rear derailleur.  All was good again.  The bearings in my freewheel started making noise last week.  Which brings me to the present.

I just replaced the rear wheel, cassette and chain on my 30-speed road bike.  The derailleurs shift perfectly and the chain travel when pedalling is smooth and quiet.  However when I stand up or dig in on a hill the chain will skip on the cassette.  It happens on all THREE chain rings in the front and the spectrum of cogs on the cassette.  I thought sure it was a stiff link in the chain, but could not find an issue with it when manually flexing each link.  I tweaked the rear derailleur to the point of shifting cogs in both directions and still have the issue.  Any suggestions?
Since you replaced the cassette and chain already, it could be your (front) chainrings that the chain is skipping on.

Check if the teeth are significantly worn (shark tooth shaped) and if the chain rides up the teeth under torque. If thats, the case, then your chainrings may need replacing.