There really is no way to tell, other then to continue to research and study it.
We all have a limited understanding of how things work, from can openers to computers, and even the universe. When a person reaches their limit of understanding regarding a particular thing, there are few ways they can respond...
1) I don't know, but I would like to know
2) I don't know, nor do I want to know (or nor do I care.)
3) I don't know, but I'm certain it's because of _______.
Lets use an automatic transmission as an example. Most people don't know how this device works, and unless they are really mechanically inquisitive, they don't want to know. As long as the car keeps moving, who cares. At the same time, most people are reasonably certain that an automatic transmission works because of the automotive engineers (who designed it) and the assembly workers (who installed it.) This explanation is acceptable to most people, and it's backed up by plenty of real-world evidence to support it. Most people would be loath to say that the reason a transmission works (or doesn't work) is because of god, because a transmission, while complex to be sure, is also comprehensible to a great many people.
Next lets take a disease, like cancer. Most people don't know how cancer works, and unless they, or someone close to them get it, they don't really care. Doctors have a pretty good handle on what causes many cancers, but plenty of laypeople believe the cause of cancer is unknown, and a few still believe it is bad luck, or punishment for evil deeds. Unlike a transmission, which is a straightforward device, cancer is alive. It can behave unpredictably, so for most people, there is an element of mystery surrounding the disease. When a terminal cancer patient beats the odds and survives, many people will see it as a miracle, or an "act of god." Doctors are far less inclined to subscribe to such outlandish explanations, mainly because they understand the mechanisms of cancer far better than the average person. They may not be able to say definitively why the cancer went into remission, but writing it off to "god" would dangerously contradict the idea that persistent research can determine the cause of disease. If doctors started down this path, they might as well go back to practicing voodoo as a way to heal.
When it comes to big unknowns, like the universe, only a minority of people in the world understand even a few of the mechanisms at work... space-time, dark matter, black holes, string theory, etc. Most people don't care why background radiation is of interest to astronomers, and unless something strongly compels us, you can pretty much guarantee that few people are sitting around trying to research the origins of the universe, figure out why time exists, or what causes gravity.
BUT, since the average persons understanding of these phenomenon is so limited, it becomes MUCH EASIER for them to write them off to god, than to try and learn or find plausible explanations. Even our educated doctors will tend to buy into the easy answer in these cases. What's more, the less people who can offer plausible explanations, the MORE people will be firmly convinced that it's "god" at work. It's an inverse effect. Socially, it is arrogant and presumptuous to announce that you understand things that nobody else in the world does, plus, having such knowledge would put you in an advantageous position relative to your fellow man, and thus you would be seen as a threat. History bears this out, as virtually all early scientists were branded dangerous heretics because they proposed that science had answers that not only did theology lack, but that theology was mistaken about.
The true answer to this question is that the one way we could tell if the universe was intelligently designed is if we figured out how to create one ourselves. Reverse engineering is probably the only way to guarantee certainty.
(ps - to some of the other posters... the statement that the universe is orderly is inaccurate. Yes, there are laws that make the universe appear orderly (thermodynamics for example), but at subatomic levels, these same laws are completely inapplicable. The universe seems orderly from our limited, human perspective, but we live on a temperate planet, in a relatively calm, orderly part of our galaxy, orbiting a stable star with a consistent period. Life on Venus would seem more chaotic, as would life were we situated near the asteroid belt, or in a galaxy that was colliding with another one, or exploding! There is plenty of disorder in the universe. You wouldn't want to experience the distortion space-time that occurs when you are caught too close to a black hole. One need only look at the cosmic debris that enters our atmosphere each night as "shooting stars" to see examples of the violent disorder spread throughout the cosmos.
(pps - awesome question, one of the best I've seen)