The majority of the movement in currencies is in the spot rates, rather than in the term structure. A 3-month rolling hedge would always be protecting against movements in the spot rates, no matter when they happen.
Using your example, if the current EUR/USD rate is 1.3333, you might be able to get a 3-month forward at 1.3339. (Forgive me if I have the direction wrong here, I haven't touched FX in years.) If the spot rate jumps to 1.2000 by the time your 3-month forward expires, you will have 0.1339 in profits to protect you from losses in your position. If you are then able to roll the hedge to a new 3-month forward at 1.2006, you still keep the 0.1339 in profits. When you finally exit you EUR position, you might only get 1.2000 for it, but you have 0.1339 in profits from that first forward. Therefore the hedge has worked, absorbing the vast majority of your currency losses.
This rolling hedge would fail if the underlying were something where the term structure was more active. Take natural gas. If you have a natural gas position -- say a producing field that will come online in an unknown number of years into the future -- then making 3-month rolling hedges would not provide much protection. Natural gas in 3 months is very uncorrelated with natural gas in 3 years.
To the extent that currency term structures do change -- meaning a move in the relative interest rates of the two countries -- you would not be protected very well. You would only be protected for the remaining life of the current 3-month forward, not for the months or years between that forward's expiration and your underlying position.
Since your old employer says that 3-month rolling hedges were the most efficient, they probably did their research. Multi-year forwards are not nearly as liquid as 3-month forwards, and they would have paid a larger bid/offer to their bank/counterparty. In the USD/EUR case they could have used Eurodollars vs. Euribor futures to hedge the interest rate risk, but the transaction costs there are not low either, and they'd have to worry about margin. They probably decided that the risk was small and the cost of a more complete hedge was high.