Mulch mowing is almost always a good thing. I used to say always, but I learned that it can cause problems in some zoysia grass lawns. I mulch mow every time I mow except if I scalp to overseed. If you mulch the clippings, you're returning nutrients as well as water and adding organic matter to the soil.

Mulch mowing is the single best thing you can do for your soil and the grass that is growing there. Often people just starting have a problem with the clippings laying on the grass and that just means you have waited too long to mow, or need to raise the mower, or are mowing too fast for the mower to work properly. Grass clippings to do what they are suppposed to do need to get down into the grass on the soil where the soil bacteria can digest them and convert them into plant food to feed the grass. You may find, especially if you apply too much Nitrogen, that your grass needs to be mowed fairly often, something I see especially in the spring when there is ample soil moisture and the grass is doing what it wants to do during the wetter and cooler times of the year, grow.
There may also be some uninformed people try to tell you that mulch mowing cause thatch which it does not.

Thank you for the responses! The things both of you said line up with my perceptions of what's going on in the lawn. I've heard some conventional-style landscapers tell me about problems with fungi and thatch, but I suspect this is mostly from experiences mulching grass that is too long.