Cleaning old paint and grime off bricks

Maybe sandblasting?

Frankly, I've never experienced anything on brick where elbow grease is better than a good pw.

We need pics for an idea of scale or what you mean by "significant". What you use depends on how well attached the stuff is. But assuming that you and I see "significant" the same way:

To start, I would pick a cloudy day (to help keep the surface nice and moist) and use a good (ie gas) pressure washer on a medium setting to slowly go at the paint and grime and then wait for it to dry to see what is left to do. If the grime is greasy and persistent, TSP mixed in a pressurized garden sprayer should help there. Let it sit for 15-30 minutes, hose it off, then pressure wash again. For stubborn excess mortar, you may need the highest setting on your pw and go at them slowly.

Make sure that you protect yourself and the garden when doing this.

Also, I would avoid sandlbasting except for extreme situations

I'll post some pics tonight. It's a mixture of what I'd call discoloration (some bricks have turned near-black in spots, for some reason), what looks like adhesive residue, paint residue that has to be several decades old at minimum, and mortar "drippings".