I'm not 100% sure, but I believe the possessive in the above sentence would be. "Jeg legger kortene mine på bordet". The possessive 'mine' is used because more than one card is being put on the table. Remember the possessive options are: min, mitt, mine. Note the original sentence is more general and not indicating possession. In terms of possession it is ambiguous; it is uncertain to whom the cards belong. If to the speaker then 'mine' could be used; if they belong to the speaker and others then 'våre'; if the cards belong to a person not present then 'sine' could be used. Hope this makes sense.
I do not think that 'I'm laying my cards on the table' would be an acceptable translation of the given sentence 'Jeg legger kortene på bordet' because the translation is more specific as it includes a reference to the possessor of the cards. However, it might be an acceptable translation of 'Jeg legger kortene mine på bordet', which is the sentence I suggested in trying to answer the questions of voxsolaus and ajosund.
A word-for-word translation wouldn't have "my" in the English. However, the associated idiom occurs much more frequently in English with a possessive than without, so I'd say that "lay my cards on the table" should be an acceptable translation.
For those interested: http://bit.ly/2cbgmln (link to a Google Ngram search; had to replace it with a shortened URL b/c Duolingo's comment system wasn't handling ampersands in query string correctly)