The first tip is to learn to listen to yourself. If someone ever corrects you, that is gold.
1) Focus on the vowels. They will help you get the resonance right (the resonance is how you hold the muscles in your mouth, so the sound seems to come from the back of your throat like in Russian or the front of your mouth like in Korean)
2) Focus on the rhythm of the language. I'm sure you know what I mean here.
3) Focus on the intonation: what pitches do they have? Scottish has kind of a lilt, for example.
4) Try to notice "phonemes", and I put it in quotes to expand the idea to mean "what native speakers consider important". For example, Mandarin speakers can't hear a vowel change to save their life, but if you get the intonation slightly wrong they glare at you.
5) Focus on details, listen to how letters change when they are between other letters, etc. (Watching to how people respond to your speech can be helpful, especially if they don't understand. Small words especially. For example, a listener might get confused and not understand the world "Lip", even though they would understand "Literally", if your "L" is slightly off)
6) Don't underestimate the value of correct grammar in improving your accent.
Incidentally, I learned of the existence of this as formal theory from David Stern (although his focus is on stage accents which is a little different)
Also, if you have any tips I'd love to hear them too :)