Is there an alternative to dissecting a real animal in the classroom?
There are many options available unfortunately however most are very expensive.
Most of the alternatives available are typically only afforded to university level (and even in some community college level) students where the school can defray the costs by passing it on to their students in the form of tuition or required course materials. Two of the most prominent alternatives to using animal cadavers are plastic models and computer generated dissection programs (virtual labs).
Plastic models are amazing, they reduce the need for formaldehyde based preservatives (reducing smell), they reduce waste, mess, and a myriad of other things that come with the dissection of animal cadavers. Unfortunately these models are very expensive, a cheap model runs around about 1,000 U.S. dollars (an expensive model can run an upwards of $15,000), a single model only gives one perspective and in a typical anatomy class uses about 30 models per semester per lab bench (I should also note that there is usually more than one section being taught at the same time, so even more models!). When it all comes out in the wash the school would be set back about $100,000 to $120,000. This cost can be easily paid-off at a university level but not so much at the high school level unless there was a significant endowment or donation to the school. I should also mention that the models break with incredible ease (as such they could last 10 years or only a semester).
Computer generated programs (virtual labs) are also great, along with the benefits that plastic models bestow upon anatomy classrooms, computer generated programs reduce the amount of time (or mitigate the need entirely) for lab instruction. Unfortunately, these programs are expensive, like $150 to $300 per student expensive. Each student would be required to pay the cost themselves (or on their parents dime), while that may not seem like much to most of the middle class, demographics vary greatly and that same $150 might not be so attainable for a family living well below the poverty line. Also many students at the high school level, at least my friends and I at the time, were/ are not yet at the maturity level where, if given a free period, they will use it as an opportunity to do what may be seen as an additional class on their own time.
Looking toward the future there is this really cool virtual reality program developed by Microsoft called the HoloLens, it's a device that you wear like glasses and it renders a 3D image that the wearer can manipulate by reaching the hand out and "moving" image or waking around the image to get a better perspective. This device is being tested by some of the U.S.'s industry leaders (education, sports, hospitals, et cetera). The HoloLens has not yet been slated for release to the public (if you're not a beta tester [read: NASA's Jet Propulsion Lab or the Cleveland Clinic] then you're just going to have to wait). Along with the unspecified amount of time the device will spend in beta testing, once the device is released to the public it will cost $3,000 per device. See the HoloLens in action in the video below, courtesy of Cleveland Clinic and Case Western Reserve University's School of Medicine:
Legal Stuff about Dissection
If you're a K-12 student in the United States and live in any of these states you can opt to have you teacher give you different assignments (instead engaging in dissections) without jeopardizing your grade. Unfortunately you're out of luck if you're no longer in K-12 (but at that point you have an option to take the course or major in biology/ kinesiology).
I hope this helps!