Voltage regulation threshold
Working on the assumption that the MCU in question will operate down to 4.7 volts or so, the simplest option is to use a LDO with a very small headroom requirement, e.g. the LP38691-ADJ/LP38693-ADJ (250 mV at 500 mA output).

Your circuit would be designed to work at 4.7 volts, and the LDO would be adjusted for this voltage. Input voltage can then range from 4.95 volts through 10 volts.

In practice, if the MCU in question can cope with Vcc of 4.5 volts, that provides you a better margin of reliability: Trim the LDO for 4.55 volts, and things will remain stable even if the input 5 volt line sags down to 4.8 volts, as it might under motors or other load. 

Another approach is to switch over to a 3.3 Volt MCU, and suitable level translation logic running off the unregulated 5-6 Volt line for those parts on the board which absolutely can not work with 3.3 volt logic. Often 5 volt logic parts accept 3.3 Volt input fine (check the datasheet), and there are 3.3 Volt MCUs (PIC18F45J10, MSP430F51x2) which are 5 Volt tolerant on selected GPIOs.