The solution is to repoint, which basically means to chip out the old mortar and replace it with new mortar. Invest in something called a "joint raker" (Buy Direct - Joint Raker). Yes, it is worth the small investment. Buy online if you have to. Scrape out loose, crumbly mortar with the joint raker. You do not have to remove all mortar, obviously, just the high spots that are loose and weakened. Then, switch to your hammer and chisel and gently pop out any stray pieces that you could not get with the joint scraper. Finish off with a wire brush to get all of the remaining crumbs out.
That was the hard part. Spray down the bricks and mortar with a garden hose, let it soak, and use your pointing trowel (Buy Direct - Pointing Trowel), which is just like it sounds: a trowel with a point on it. Press in the new mortar, matching existing good mortar as much as possible.
Here is the caveat: repointing mortar is fun for the first six to eight bricks. After that, it's just plain hard work. Take it slow, set small goals for yourself every day, and set your mind on getting that whole wall done over a course of weeks or months, instead of a weekend.