I have been listening to it a lot recently and everytime I've thought: what a great album.
I listened to it a lot recently and I thought: What a great album.
Which sentence is better? I think the first one is.
I've been listening to it a lot....
No need to say "recently" with this tense, as it already means recently.
I've listened to it a lot recently...
We need "recently" here if the listening has been recent (because it could mean "over the course of my life" without a more specific time reference).
I listened to it a lot recently... [not (quite) OK]
The simple past "listened" and the adverbial "a lot" and "recently" are discordant. (But you will hear more than a few native speakers say it this way and it's just a little jarring to the ears of those speakers who tend to use tenses in a more nuanced manner.)