So I was listening to a radio sermon on the Roman guard who, according to Scripture, was set to guard the tomb of Christ.
So, this isn't a religious question, but the preacher was mentioning that the Roman Legionnaire's were the most disciplined soldiers of the day. If they fled from battle, the punishment was execution. If they fell asleep during a night watch, the punishment was execution.
Is this true? Were there any other (what we would consider) harsh punishments?
Though I can't speak to them being the most disciplined of their time, observers such as Josephus (A History of the Jewish War) commented that they were noticeably more trained and organised than the norm.
As for punishments, the Roman's were pretty harsh by our standards. Polybius (History of the World) writes that punishment for falling asleep during a night watch is being bludgeoned to death with sticks and stones by the whole camp. Additional crimes deserving of bludgeoning are stealing from the camp, bearing false witness, abuse of one's own body, making false claims of valour to one's tribune, abandoning an assigned post in the face of an enemy (Modestinus writes that this was also punished by torture and death by wild animals or the rack), dropping one's arms in fear during a battle, violating any direct order from a superior in battle, and being fined three times for the same offence.
In the case that an entire group (century, maniple, etc) fled battle, once they were recaptured, the entire legion would be assembled and the guilty parties marched to the center of the assembly. There the tribune would castigate them for their offence, bring forth the first man observed to flee, and execute him in front of the group. The officers would then chose by lottery the names of ten percent of the deserters and have them executed by being bludgeoned to death (notice that Romans liked bludgeoning yet?) by their fellows, which is where the word "decimation" or "the reduction by one tenth" comes from. The remaining deserters would then be put on a ration of barley instead of wheat and forced to sleep outside the protection of the camp's walls.
(Translations by Shelton)
TL;DR - Yes, the Romans practiced harsh punishments like bludgeoning and flogging, were noticeably more disciplined than their contemporaries, and punished desertion with more bludgeoning (Note that corporal punishment for desertion is a fairly common treatment regardless of group).
(Edits since I haven't slept and fail at formatting)
The Roman army was arguably one of the most sophisticated and modern (in terms of thinking) armies of the ancient world. In the Augustan period and beyond, the army was a career choice by the people, rather than being a civic duty you were called up for. This generally created a class of citizen with motivations above mere survival and "making it" through their military service.
When a Roman joined the army he swore an oath (the sacramentum militae) pledging their loyalty to their consul (in the republican period) or the emperor. The oath was basically the soldier saying "if I neglect my duty I will not object to being killed as punishment" and was not unlike the same oath sworn by the gladiators who committed themselves to be:
Burned, bound beaten and put to the sword should the possibility arise (Sen. Epist. 71:32)
As the men were sworn in for life and took such risks to join up, the army often had great rewards for valour in battle and service not unlike the medals system modern armies employ:
After a battle in which some of them have distinguished themselves, the general the troops together, and brings forward those whom he considers to have displayed great valour (Polyb. III. 39).
The rewards included torcs, armbands and replicas of the legionary standards among others.
As for the punishments, /u/boromakot has covered this really well, though there are some cases where soldiers would be just dishonorably discharged. As you can tell, the punishments for dereliction of duty were very severe and I'm sure many Legionaries knew not to break the rules.
My own interpretation, however, especially when reading of acts committed during the Punic Wars and Macedonian Wars (my area of expertise is based in republican Rome, rather than imperial) is that the ethos that Roman society imbued about war meant that many men were proud of serving in battle and in the army. It was a manly thing done for one's country and the Romans habitually mobilised 18 to 24 per cent of their population every year (Harris, 1979, 42-44).
Not that this hasn't stopped rebellious acts through Roman history. An army of veterans were on the brink of mutiny during the Second Macedonian War due to not receiving (in their minds) enough booty from the campaign. Only a timely victory and a haul of spoils stopped them from refusing to fight under their commanders.
BTW, it can be debated if the guard at the tomb of Jesus (attested in only one of the gospel, that of Matthew) was composed by Roman Legionnaires under the command of Pilate or by members of a local force under the command of the temple authorities.
After the resurrection they make their report to the chief priests, not to Pilate (Matthew 28, 11-15).