I believe that you are not really asking about "The Flea." The phrase "country pleasures" does not appear in that poem. However, it does appear in "The Good-Morrow." In that poem, the phrase is referring to pleasures that are really not sophisticated. They are pleasures that country people would enjoy but that more "civilized" people would look down on.
This really goes with the idea of this poem as a whole. The speaker is saying that he and his love were nothing until they fell in love. He compares them to children who hadn't grown up. In the phrase you cite, he compares them to country people who had not really become civilized or sophisticated.
So I believe the phrase refers to things that only unsophisticated "hicks" would find enjoyable.
The phrase 'country pleasures' occurs in the first verse of Donne's poem, The good-Morrow:
I wonder, by my troth, what thou and I
Did, till we loved? were we not weaned till then,
But sucked on country pleasures, childishly?
In this poem, the lover addresses the beloved in the first person mode with a question which is followed by more questions that seem to contain answers to the opening one. The phrase 'country pleasures' is part of the questionnaire.
In the three verses of the poem, the speaker initiates a journey, a journey from the past to the present to the future of love. a journey from the body to the mind and thence to the spirit. The opening question looks back into their past history of love, a sheer bodily engagement which they mistook for love. They were actually separated while they were under the illusory impression of union. 'were we not weaned till then?' is the question that looks back into that state. 'But sucked on country pleasures childishly?' extends the interrogation. Despite being adult persons, the two lovers behaved 'childishly' and wanted to enjoy bodily pleasures. By the phrase 'country pleasures', the lover means the gross rustic pleasures of the senses when the souls were in a state of slumber.
One of the several possible meanings to the phrase "country pleasures" is its use as a slang phrase, used euphemistically to refer to sexual intercourse.