The general FAA regulation on turbine engines on airliners is § 25.939
§ 25.939 Turbine engine operating characteristics.
(a) Turbine engine operating characteristics must be investigated in flight to determine that no adverse characteristics (such as stall, surge, or flameout) are present, to a hazardous degree, during normal and emergency operation within the range of operating limitations of the airplane and of the engine.
(c) The turbine engine air inlet system may not, as a result of air flow distortion during normal operation, cause vibration harmful to the engine.
The FAA has published Advisory Circular 25.939-1 which provides more detail as to how this regulation should be met. The takeoff phase is critical for safety, so the FAA requires testing to show that:
No adverse engine operating characteristics (mild, moderate, or severe) should exist after the power setting phase (normally completed by 60 to 80 knots) of the takeoff procedure through attainment of the enroute configuration and climb to 1,500 ft. above the airport.
The "adverse operating characteristics" are defined in more detail earlier in the AC. So to answer the question, no, it is required by law that jet engines have no issue with takeoffs in crosswinds. The faster an aircraft is moving, the less the crosswind airspeed component is. Once the aircraft is flying, aside from control inputs or changes in the atmosphere, the air stream comes from straight ahead.
The highest amount of relative crosswind will actually occur while the aircraft is sitting on the ground. Note that unlike the takeoff phase:
Taxiing: No adverse engine operating characteristics (mild, moderate, or severe) should exist during taxiing except for operation in crosswinds and tailwinds where mild adverse operating characteristics are acceptable.