There are, technically, no problems with building a pseudo-Orion using conventional explosives - that is, in fact, what they did to test the concept. The practical problem, as alluded to by Joe Bloggs' answer, is the difference in energy density between conventional and nuclear explosives.
Part of the reason that, say, automotive gasoline and oxygen aren't (usually) used as rocket fuel (they prefer an ultra-refined kerosene) - is that in addition to fouling problems, the specific impulse of gasoline isn't spectacular. (This isn't to say it can't be done, though.)
However, all of this assumes that you're using a rocket nozzle to optimize the speed of your propellant stream. Orion didn't do that; it used a pusher plate, and (this is actually pretty cool to be able to say) a shaped nuclear charge (diagrammed here). The pusher plate (an enormous, metres-thick slab of steel connected to the ship proper by equally-enormous shock-absorbers) had the fast-moving propellant, now a plasma thanks to the nuclear detonation, hit it, absorbed the energy and let Newton do the rest. This allowed for an enormous amount of thrust, but wasn't nearly as efficient as a rocket engine. And here's where we get to the math bit.
The theoretical maximum achievable efficiency of a pure-fission device (because you wouldn't want to be using fusion weapons with a Project Orion unless it was really big) is about 0.1kt per kg of mass (source). But this means that you get the explosive force of 100 000kg of TNT for every 1kg of bomb weight. Assuming that packaging and propellant are a constant and can be disregarded, that's still a lot of bang for your buck, which is extremely important in rocketry. The most powerful known chemical explosive is Octanitrocubane, which has an RE factor of 2.38, making it 2.38 times as powerful as TNT per unit mass. Pretty good, but you'd still need forty thousand times as much mass of fuel to achieve the same amount of thrust, and rocketry doesn't allow for that The more mass you require to achieve the same amount of thrust, the less you can lift with it, and it's non-linear (since you have to bring the mass with you).
The Tsiolkovsky rocket equation applies here, though it's difficult (unless you're a physicist specialising in bomb design) to determine the exact specific impulse of a shaped nuclear charge's plasma propellant impacting a pusher plate. Basically, the higher the specific impulse (which is probably associated, when using explosives, more or less linearly with the explosive force generated by the bomb), the greater the initial mass of the rocket can be (assuming you're starting from a gravity well). If your efficiency (explosive yield per unit mass) is too low, you can't lift as much.
So, to sum this up, if you're not using nukes, you can't use the Orion design to make a useful rocket, let alone a less expensive option.
- You could probably get a little better yield-per-mass from your nuclear weapons with boosted fission designs, but I stuck with pure fission weapons because that's what they were working with in 1966 when they were designing Orion.
- Just because nukes were off the table didn't mean that rocketry scientists weren't up to trying everything; if a conventional-explosive bomb-driven rocket were actually in any way more efficient or cheaper, you can bet that they'd already have them in action. They aren't... so they aren't.
- Pursuant to #2, I recommend giving Ignition! by John D. Clark a read. When I say rocket scientists (and rocket chemists in particular) were up to trying everything, I mean everything. It was nuts. And it's a great read!
Edit: Loren Pechtel's figures are right, I was mistaking kt for t. Tens of thousands of kg of TNT per kg of bomb, rather than hundreds. (This actually makes the conventional-explosive variety of Orion less plausible.)