What are some good games NOT from Konami I haven’t played yet?

I love Fray: good music, voices in Turbo-R version. And you can decrease the speed of the game if it's too complicated.
I also used to play RPG, one of my favourite is Tower of Gazzel.
And the tetris-like game Puyo-puyo.

Hydefos, a hori shmup, has great music and it's good fun. Feedback, a third person rail shooter, is great. And Strategic Mars, a space combat strategy game, can be surprisingly engrossing (though its clunky gameplay may be difficult to endure nowadays). Herzog, a real-time strategy game, despite being slow, is ahead of its time. Of course a lot of stuff from Falcom and Compile is also great.

This is a bit odd question - for making recommendations for that question it would be needed to know what have you played so far

Super Rambo, Ashguine 2, ZanacEx, Tengoku Yoitoko

Hmmm.... *looks at shelf*

Compile games - Guardic, Golvellius, Shin Golvellius, Higemaru Makaijima, Girly Block, Nyanpi, Gorby's Pipeline

Andorogynus, Mick and Rick, Mon Mon Monster, Quinpl, Vaxol, War of the Dead Part 2 with translation patch, Yurei-kun

Nearing unobtainium - Psycho World, Pleasure Hearts (I've seen this reasonably priced, but only for sale once and it was gone before I found it), Star Virgin, Alpha Roid (some people really like this game, I don't see it myself), Yurei-kun is borderline this category but it comes up for sale once or twice a year, just really pricey, Goblin, Come on Picot!