CE-680A Captain reported a near miss with another aircraft departing on a crossing runway at a non-towered airport.

Narrative

We conducted a normal flight into this airport in order to pick up our owner on a fairly routine flight. On taxi out to depart runway XX; we had another small single engine aircraft that was departing on a crossing runway (YR). That traffic reported departing YR; and did so normally. We reported on CTAF that we were departing runway XX; and conducted the takeoff static; but normally. Once airborne I noticed that the departing aircraft was starting a RIGHT turn directly in front of us into the departure corridor of our runway. I adjusted my climb profile to ensure that I had at least 1000 feet clearance above the traffic that turned in front of us; but it was a surprise that the traffic did so. This was a lot closer separation than I would normally be comfortable with. Runway XX is listed as the runway preferred use runway due to noise and weight restriction.Suggestions: This airport needs to be reviewed immediately for safety of operations! The departing traffic may or may not broadcast that they are staying in the pattern; which is right traffic off of a crossing runway.The runways in the airport are VERY short; and after reviewing the notes for this airport; it appears that we should also be doing noise abatement takeoffs; which is not conducive to safe operation out of short runways. (Especially with the increased chances of traffic turning right into the departure corridor of our preferred departure runway for noise/weight.)