For example, an elemental gem or other device (Bowl of Commanding Water Elementals, etc.) that summons an elemental: Do you then use up your concentration on that summoning, even though you used a magic item to do the summoning? In this case, it would allow you the spell but not benefit you in terms of allowing other spells while it is in effect.

If the item let's you cast a spell with concentration, you are considered to be concentrating on the spell unless the item says otherwise. For example, sentient items can concentrate on spells they cast

What Dave said. If you activated the effect as if casting a spell, then you are concentrating on the effect as if concentrating on the spell.

You are both clear . . . and yet I think there is still a question. So if I throw down that elemental gem, it splits and summons an elemental...that's not "like casting a spell." THe description is, "When you use an action to break the gem, an elemental is summoned as if you had cast the Conjure Elemental spell, and the gem's magic is lost." That does not sound like you have to keep concentrating. Yet it says "as if you had cast..." which would imply concentration. 

The Bowl of Commanding Water Elementals seems a little clearer. "While this bowl is filled with water, you can use an action to speak the bowl's Command Word and summon a Water Elemental, as if you had cast the Conjure Elemental spell..." That sounds like, well, you cast the spell. So I might expect concentration be needed to keep it around.

They both use the same "as if you had cast" and that is the most important part. If it says that - you're the one who cast the spell - even if the fluff text doesn't lend itself to that image. So in both cases you do need to concentrate.

Magic items state when concentration is specifically not required.  For example, compare the Potion of Clairvoyance with the Potion of Speed.