My doctor said I had UTI symptoms and prescribed antibiotics without performing a urine culture. Should I be concerned?

Depends. Usually if you are not pregnant and have classic symptoms of a urinary tract infection, then you don't need to do a urine culture. You should take a simple antibiotic for three days. If you are pregnant, however, a urine culture should be obtained prior to initiating antibiotcs. You should also have a repeat urine culture after you have finished the antibiotics to make sure the infection is gone.
Hopefully urinalysis. Minimum standard of care would require at least a simple dipstick urine analysis to be performed, this would be nitrite positive if bacteria were present & probably leucocyte esterase positive if white blood cells were present. Many drs do not order a urine culture for an isolated UTI with typical symtoms. It certainly reduces management costs. Some drs always get urine cultures.
Not necessarily. Generally, it is better to have a culture in order to identify the bacteria and ensure that it is sensitive to the antibiotic chosen. It is possible for non-infectious problems to cause symptoms similar to a uti. If the symptoms do not resolve after a course of antibiotics you should probably return for a urine culture prior to further treatment.