You can make a Diplomacy check to gather information, asking people nearby if anyone saw someone near your dining room. You can make a Sense Motive check as you walk around and see if anyone acts suspicious when you bring the topic up.
If the trail is recent, you could get an animal to do scent tracking. The DC for this is 10 for a very recent trail, +2 per hour the trail is old. If you had a scroll of lesser planar ally, you could summon a creature such as a foo dog which speaks common and has the scent special ability. If you can find someone with a trained bloodhound, that would work just as well.
The commune spell (spell level five) allows you to ask your deity yes-or-no questions. The divination spell (spell level four) might also give useful information.
Your last option is to accept that someone's probably going to try to attack you in the near future, and have your group take reasonable precautions against attacks, and expect to identify the attacker when they do the attack. This is not very satisfying from a narrative standpoint, but it's the most common outcome when an adventuring party receives a note like this, assuming the attacker took reasonable precautions against getting discovered. (Presumably the attacker thinks they've taken reasonable precautions against getting discovered, so all the approaches above are essentially hoping the attacker has made a foolish error.)
Also, consider that your DM might be more interested in the story outcome where you get ambushed and then defeat them in a dramatic battle. If your DM is nice, they might allow you to generate a different outcome if you push for it hard enough. But this might not be the be the best use of your effort.