I've been replanting my garlic every year from some seed cloves I purchased 10 years ago. Every year, I replant cloves from the largest, nicest heads and use the smaller ones in the kitchen. Despite this, I've noticed that I get fewer and fewer nice, large heads every year. Is there a better way to select the best heads for replanting so the size of my garlic doesn't decrease over time?
A few thoughts that might be germane to your situation.
If this is a hard-necked variety of garlic, do you cut the scape (the flower stalk that shoots up) off of it? That will produce a larger bulb - at least that has been my observance of the situation. The theory is that cutting off the scape allows (forces?) the plant to put its energy into bulb development rather than in growing that flower stalk.
What about soil amendment? Over the years are you amending the soil where you plant your garlic? I've seen a significant difference in the health and size of my garlic (and all plants, really) when I've focused on soil health. Garlic, like many plants, is a heavy feeder and needs a healthy soil. And check the pH of the soil. I think you're going to want to shoot for something in the 6.5 - 7.0 range. Plants like garlic will grow in less than ideal soil but getting the pH right is a good step in getting them to thrive.
Do you rotate the garlic between beds or always plant in the same place? I believe that crop rotation is important and think that a good rule of thumb is that you don't plant garlic (or anything planted annually) in the same place two years in a row. If space is available, consider rotating the garlic among maybe 3 locations. My rotational plan is to move everything around yearly and end up back at square one after 3 or 4 years. This may or may not be feasible depending upon the space you have available.
Generally speaking, I think your choosing of the best bulbs is the right approach - it is time-tested and encourages those traits in the future. Not knowing anything more about your particular situation that what's in your question, I'd guess that it's likely the soil/soil conditions that are the culprit. That's generally been my conclusion here on our farm when plants perform below my expectations.
A few observations from a long-time garlic grower (just harvested today, actually): You may simply have varieties that don't do all that well in your microclimate - I finally gave up on the garlic I had been growing (which was diminishing noticeably over time) and bought some new, known seed (Spanish Roja) a few years back, and also got a seed head from someone else that had heirloom (grandma's) garlic that was doing better here.
3" apart is WAY too close, IMPE. Mine are 8" apart and I might go 9" next year. Perhaps you are in rows, but even so, way too close (mine are grid-planted 8x8, considering 9x9)
Large bulbs is good. Saving enough large bulbs that you can plant only the large cloves from the large bulbs is better - eat the smaller ones. Resulting head size has been tied to clove size more strongly than head size. I've finally gone and tracked down some links (and then one died, so here's one link, anyway): http://hortsci.ashspublications.org/content/39/6/1272.full.pdf+html
Nutrition counts. The "new heirloom" variety came to me as an impressively large bulb. In the second year I've grown it, it has large bulbs that match the Spanish Roja, but overall the bulbs are smaller - it happened to get the less-well-treated of the two beds, and it showed at harvest.
My personal experiments with scaping/not scaping lead me to scape everything this year as there was a small but noticeable effect on size. Some literature does suggest that leaving a few scapes and growing out seed bulbils from the top may be beneficial to break some soil-borne disease cycles, but it takes a year or two to get from scape bulbil to regular head of garlic (possibly longer with varieties that make smaller scape bulbils.)
I also used to wait until most of the leaves were brown. My heads never had a proper wrapper. This is related. I now try to harvest when 3-4 leaves are still green (often tinged with brown) which results in more of a wrapper on the head, and less dirt inside. If collecting scape bulbils, literature claims you should let them go longer (i.e., full brown) to help mature the bulbils.
What is your depth of planting? I used to put mine way too deep - recommendations are pretty much twice the depth of the clove, and paying attention to the clove orientation (base down, tip up.)