Straw serves 2 purposes. To retain moisture 2. Preventing wash-out or erosion. If you have a system you have moisture on demand. It still may be a good idea to use straw on uneven or sloping contours to prevent erosion.
Straw can prevent needed sunlight from reaching new seedlings, therefore inhibiting growth.
At first, water your new seed 2 or 3 times daily during the daylight hours for brief periods. Watering once in the early morning for a longer period. During this initial period, avoid walking on the yard. After about a month, your new lawn may be ready to mow for the first time. You'll have to judge this based on it's growth.