I own and frequently wear a navy cotton unlined sport coat from cp company, as well as a half lined tan sport coat that I picked up for $40 at Daffy's. I can get through a week in the office during the summer with these if needed because of their basic coloring. The unlined aspect of them makes a world of difference. I own the j crew linen suit in navy, and while I like the look, it wears very warm as it is a fully lined jacket.
I had Hemrajani make up a 1/3rd lined sport coat from a Loro Piana linen and silk fabric last year which is pretty incredible. Far too busy to wear to the office, it's my default vacation blazer and acks well.
Just picked up a navy cotton SC from Banana Republic that I really like. It's a tad on the heavy side for a summer SC, but I think it's very versatile nonetheless. I know BR isn't loved around here, but it works well casually and you can dress it up with a tie and nice trousers pretty easily.
I also have this linen one by Luciano Barbera also with patch pockets. Cream linen is wonderful for an occasion like a wedding or a party but is seems a tad precious for everyday wear so it just doesn't get a lot of use. I will get a tan or brown(ish) linen suit next time.