Captive garter snakes do not have the same resistance to disease and parasites that wild garter snakes do, and a wild garter's diet is chock-full of parasites that they normally tolerate well enough. Fish, frogs and earthworms all contain internal parasites -- but then so do wild mice. Feeding wild mice to a captive-bred corn snake is an awfully good way to get it sick; it's the same principle when feeding a garter's traditional diet to a garter. The difference is that there isn't the same availability of clean food items -- i.e., there are no fish/frog equivalents of lab/domestic mice.
So what do you do? As Flo and others pointed out, feeding live fish can cause all sorts of problems -- even if the fish doesn't have the thiaminase enzyme that destroys Vitamin B1. Feeder fish are chock-full of parasites, and that goes for goldfish, too, which are nutritionally junk fish at that. Frozen fish from the supermarket doesn't have the same problems, except that it's not nutritionally complete: the fillet doesn't contain the nutrients that the head and guts would. A diet of ocean perch, while inexpensive and safe from the point of view of thiaminase and internal parasites, wouldn't be a good idea.
Earthworms are usually a pretty good bet, except that the places they're collected from -- golf courses, for example -- are hardly examples of clean, pesticide-free locales. So you're running into risks there as well. Now I've used bait-store nightcrawlers with no apparent problems, but I know full well that those problems may present themselves down the line.
When you consider all the problems inherent in using fish and worms -- and frogs, with their parasite issues, which would otherwise be ideal -- you can see why many garter snake keepers opt for mice. Mice are nutritionally complete, as any corn snake or kingsnake keeper will tell you. And mice are vertebrates, just like frogs, so to say that a garter snake's system can't handle them is overstating things just a tad, and isn't really an informed opinion. Even earthworm specialists like Butler's garter snakes will take mice happily --
http://www.mcwetboy.com/articles/butlers.phtml -- and it's been documented that the diet of adult female garter snakes on the prairies -- wandering, plains, red-sided garters -- is almost exclusively rodents. They need the nutrition.
Finally, all I can say is to look at the experience of garter keepers who've been at it for a while. I've had seven of my garter snakes between 2.5 and three years on an almost exclusive mouse diet: eastern, red-sided, wandering and Butler's garters. They're doing really well. The snakes that were harder to convert to mice -- a melanistic eastern and an eastern blackneck -- ate mostly fish and have since died.
As for converting, almost all of my wandering, red-sided and eastern babies took UNSCENTED pinky parts the first time they were offered. In many cases, it depends on the species or locality from which they or their parents were collected -- some are just more die-hard fish-eaters than others.
Here's my care sheet again:
http://www.mcwetboy.com/gartercare/